Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

ABDULLAYEV F., Özkartepe P. N.

Science China Mathematics, vol.58, no.5, pp.1063-1078, 2015 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 58 Issue: 5
  • Publication Date: 2015
  • Doi Number: 10.1007/s11425-014-4908-x
  • Journal Name: Science China Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1063-1078
  • Keywords: Bieberbach polynomials, complex approximation, conformal mapping, quasiconformal curve
  • Uşak University Affiliated: No

Abstract

Let G ⊂ ℂ be a simply connected domain whose boundary L := ∂G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| < r0}, satisfying φ(0) = 0, φ′(0) = 1. We consider the following extremal problem for p > 0: (Formula Presented.) in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, Pn′ (0) = 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the φ(z) on (Formula Presented.) with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their “touch”.